October  2017, 22(8): 3199-3220. doi: 10.3934/dcdsb.2017170

Exponential trichotomy and $(r, p)$-admissibility for discrete dynamical systems

1. 

Faculty of Mathematics and Computer Science, West University of Timişoara, Pârvan Blvd. No. 4,300223 Timişoara, Romania

2. 

Academy of Romanian Scientists, Splaiul Independenţei 54,050094, Bucharest, Romania

Received  February 2016 Revised  October 2016 Published  June 2017

The aim of this paper is to present a new and very general method for the study of the uniform exponential trichotomy of nonautonomous dynamical systems defined on the whole axis. We consider a discrete dynamical system and we introduce the property of $(r, p)$-admissibility relative to an associated control system, where $r, p∈ [1, ∞]$. In several constructive steps, we obtain full descriptions of the sufficient conditions and respectively of the necessary criteria for uniform exponential trichotomy based on the $(r, p)$-admissibility of the pair $(\ell^∞(\mathbb{Z}, X), \ell^1(\mathbb{Z}, X))$. In the same time, we provide a complete diagram of the $\ell^p$-spaces which can be considered in the admissible pairs for the study of the uniform exponential trichotomy of discrete dynamical systems. We present illustrative examples in order to motivate the hypotheses and the generality of our method. Finally, we apply the main results to obtain new criteria for uniform exponential trichotomy of dynamical systems modeled by evolution families using the admissibility of various pairs of $\ell^p$-spaces.

Citation: Adina Luminiţa Sasu, Bogdan Sasu. Exponential trichotomy and $(r, p)$-admissibility for discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3199-3220. doi: 10.3934/dcdsb.2017170
References:
[1]

C. V. Coffman and J. J. Schäffer, Dichotomies for linear difference equations, Math. Ann., 172 (1967), 139-166.  doi: 10.1007/BF01350095.  Google Scholar

[2]

S. Elaydi and O. Hájek, Exponential trichotomy of differential systems, J. Math. Anal. Appl., 129 (1988), 362-374.  doi: 10.1016/0022-247X(88)90255-7.  Google Scholar

[3]

S. Elaydi and O. Hájek, Exponential dichotomy and trichotomy of nonlinear differential equations, Differ. Integral Equ., 3 (1990), 1201-1224.   Google Scholar

[4]

S. Elaydi and K. Janglajew, Dichotomy and trichotomy of difference equations, J. Differ. Equations Appl., 3 (1998), 417-448.  doi: 10.1080/10236199708808113.  Google Scholar

[5]

J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Academic Press, 1966.  Google Scholar

[6]

N. Van Minh, On the proof of characterisations of the exponential dichotomy, Proc. Amer. Math. Soc., 127 (1999), 779-782.  doi: 10.1090/S0002-9939-99-04640-7.  Google Scholar

[7]

N. Van Minh and N. T. Huy, Characterizations of dichotomies of evolution equations on the half-line, J. Math. Anal. Appl., 261 (2001), 28-44.  doi: 10.1006/jmaa.2001.7450.  Google Scholar

[8]

N. Van Minh and J. Wu, Invariant manifolds of partial functional differential equations, J. Differential Equations, 198 (2004), 381-421.  doi: 10.1016/j.jde.2003.10.006.  Google Scholar

[9]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256.  doi: 10.1016/0022-0396(84)90082-2.  Google Scholar

[10]

K. J. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156.  doi: 10.1090/S0002-9939-1988-0958058-1.  Google Scholar

[11]

K. J. Palmer, Shadowing and Silnikov chaos, Nonlinear Anal., 27 (1996), 1075-1093.  doi: 10.1016/0362-546X(95)00042-T.  Google Scholar

[12]

O. Perron, Die Stabilitätsfrage bei Differentialgleischungen, Math. Z., 32 (1930), 703-728.  doi: 10.1007/BF01194662.  Google Scholar

[13]

V. A. Pliss and G. R. Sell, Robustness of the exponential dichotomy in infinite-dimensional dynamical systems, J. Dynam. Differential Equations, 3 (1999), 471-513.  doi: 10.1023/A:1021913903923.  Google Scholar

[14]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems Lecture Notes in Mathematics, vol. 2002, Springer, 2010. doi: 10.1007/978-3-642-14258-1.  Google Scholar

[15]

C. Pötzsche, Fine structure of the dichotomy spectrum, Integral Equations Operator Theory, 73 (2012), 107-151.  doi: 10.1007/s00020-012-1959-7.  Google Scholar

[16]

C. Pötzsche, Dichotomy spectra of triangular equations, Discrete Contin. Dyn. Syst., 36 (2016), 423-450.  doi: 10.3934/dcds.2016.36.423.  Google Scholar

[17]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems, III, J. Differential Equations, 22 (1976), 497-522.  doi: 10.1016/0022-0396(76)90043-7.  Google Scholar

[18]

B. Sasu and A. L. Sasu, Exponential dichotomy and $(\ell^p, \ell^q) $-admissibility on the half-line, J. Math. Anal. Appl., 316 (2006), 397-408.  doi: 10.1016/j.jmaa.2005.04.047.  Google Scholar

[19]

B. Sasu, Uniform dichotomy and exponential dichotomy of evolution families on the half-line, J. Math. Anal. Appl., 323 (2006), 1465-1478.  doi: 10.1016/j.jmaa.2005.12.002.  Google Scholar

[20]

B. Sasu and A. L. Sasu, Exponential trichotomy and p-admissibility for evolution families on the real line, Math. Z., 253 (2006), 515-536.  doi: 10.1007/s00209-005-0920-8.  Google Scholar

[21]

A. L. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl., 344 (2008), 906-920.  doi: 10.1016/j.jmaa.2008.03.019.  Google Scholar

[22]

A. L. Sasu and B. Sasu, Input-output admissibility and exponential trichotomy of difference equations, J. Math. Anal. Appl., 380 (2011), 17-32.  doi: 10.1016/j.jmaa.2011.02.045.  Google Scholar

[23]

B. Sasu and A. L. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line, Discrete Contin. Dyn. Syst., 33 (2013), 3057-3084.  doi: 10.3934/dcds.2013.33.3057.  Google Scholar

[24]

A. L. Sasu and B. Sasu, Discrete admissibility and exponential trichotomy of dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 2929-2962.  doi: 10.3934/dcds.2014.34.2929.  Google Scholar

[25]

A. L. Sasu and B. Sasu, Admissibility and exponential trichotomy of dynamical systems described by skew-product flows, J. Differential Equations, 260 (2016), 1656-1689.  doi: 10.1016/j.jde.2015.09.042.  Google Scholar

show all references

References:
[1]

C. V. Coffman and J. J. Schäffer, Dichotomies for linear difference equations, Math. Ann., 172 (1967), 139-166.  doi: 10.1007/BF01350095.  Google Scholar

[2]

S. Elaydi and O. Hájek, Exponential trichotomy of differential systems, J. Math. Anal. Appl., 129 (1988), 362-374.  doi: 10.1016/0022-247X(88)90255-7.  Google Scholar

[3]

S. Elaydi and O. Hájek, Exponential dichotomy and trichotomy of nonlinear differential equations, Differ. Integral Equ., 3 (1990), 1201-1224.   Google Scholar

[4]

S. Elaydi and K. Janglajew, Dichotomy and trichotomy of difference equations, J. Differ. Equations Appl., 3 (1998), 417-448.  doi: 10.1080/10236199708808113.  Google Scholar

[5]

J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Academic Press, 1966.  Google Scholar

[6]

N. Van Minh, On the proof of characterisations of the exponential dichotomy, Proc. Amer. Math. Soc., 127 (1999), 779-782.  doi: 10.1090/S0002-9939-99-04640-7.  Google Scholar

[7]

N. Van Minh and N. T. Huy, Characterizations of dichotomies of evolution equations on the half-line, J. Math. Anal. Appl., 261 (2001), 28-44.  doi: 10.1006/jmaa.2001.7450.  Google Scholar

[8]

N. Van Minh and J. Wu, Invariant manifolds of partial functional differential equations, J. Differential Equations, 198 (2004), 381-421.  doi: 10.1016/j.jde.2003.10.006.  Google Scholar

[9]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256.  doi: 10.1016/0022-0396(84)90082-2.  Google Scholar

[10]

K. J. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156.  doi: 10.1090/S0002-9939-1988-0958058-1.  Google Scholar

[11]

K. J. Palmer, Shadowing and Silnikov chaos, Nonlinear Anal., 27 (1996), 1075-1093.  doi: 10.1016/0362-546X(95)00042-T.  Google Scholar

[12]

O. Perron, Die Stabilitätsfrage bei Differentialgleischungen, Math. Z., 32 (1930), 703-728.  doi: 10.1007/BF01194662.  Google Scholar

[13]

V. A. Pliss and G. R. Sell, Robustness of the exponential dichotomy in infinite-dimensional dynamical systems, J. Dynam. Differential Equations, 3 (1999), 471-513.  doi: 10.1023/A:1021913903923.  Google Scholar

[14]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems Lecture Notes in Mathematics, vol. 2002, Springer, 2010. doi: 10.1007/978-3-642-14258-1.  Google Scholar

[15]

C. Pötzsche, Fine structure of the dichotomy spectrum, Integral Equations Operator Theory, 73 (2012), 107-151.  doi: 10.1007/s00020-012-1959-7.  Google Scholar

[16]

C. Pötzsche, Dichotomy spectra of triangular equations, Discrete Contin. Dyn. Syst., 36 (2016), 423-450.  doi: 10.3934/dcds.2016.36.423.  Google Scholar

[17]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems, III, J. Differential Equations, 22 (1976), 497-522.  doi: 10.1016/0022-0396(76)90043-7.  Google Scholar

[18]

B. Sasu and A. L. Sasu, Exponential dichotomy and $(\ell^p, \ell^q) $-admissibility on the half-line, J. Math. Anal. Appl., 316 (2006), 397-408.  doi: 10.1016/j.jmaa.2005.04.047.  Google Scholar

[19]

B. Sasu, Uniform dichotomy and exponential dichotomy of evolution families on the half-line, J. Math. Anal. Appl., 323 (2006), 1465-1478.  doi: 10.1016/j.jmaa.2005.12.002.  Google Scholar

[20]

B. Sasu and A. L. Sasu, Exponential trichotomy and p-admissibility for evolution families on the real line, Math. Z., 253 (2006), 515-536.  doi: 10.1007/s00209-005-0920-8.  Google Scholar

[21]

A. L. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl., 344 (2008), 906-920.  doi: 10.1016/j.jmaa.2008.03.019.  Google Scholar

[22]

A. L. Sasu and B. Sasu, Input-output admissibility and exponential trichotomy of difference equations, J. Math. Anal. Appl., 380 (2011), 17-32.  doi: 10.1016/j.jmaa.2011.02.045.  Google Scholar

[23]

B. Sasu and A. L. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line, Discrete Contin. Dyn. Syst., 33 (2013), 3057-3084.  doi: 10.3934/dcds.2013.33.3057.  Google Scholar

[24]

A. L. Sasu and B. Sasu, Discrete admissibility and exponential trichotomy of dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 2929-2962.  doi: 10.3934/dcds.2014.34.2929.  Google Scholar

[25]

A. L. Sasu and B. Sasu, Admissibility and exponential trichotomy of dynamical systems described by skew-product flows, J. Differential Equations, 260 (2016), 1656-1689.  doi: 10.1016/j.jde.2015.09.042.  Google Scholar

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